Exactly on of the following alternatives holds.
Either the inequality $Ay\ge c \ \ (1)$ has a solution or the $xA=0$, $xc=1 \ \ (2)$ has a nonnegative solution. $A$ is matix of size $m$ x $n$, $x$ an $c$ are vectors of size $m$, $y$ is vector or size $n$.
I have showed that $(1)$ and $(2)$ cannot have solutions simultaneously. How to show if one of them has not solution the other has?